The conserved quantity arising from a rotational invariance. Combine with rotational-dynamics for the classical mechanics approach and quantum-mechanics for the QM interpretation
-1
votes
1answer
110 views
Can an electric motor force angular momentum not to be conserved in an isolated system?
An ice skater is in a spin, she pulls her arms in and she spins faster, she lets her arms extend outward and then she starts to slow down.
She will probably weigh on a weigh scale about the same ...
2
votes
1answer
371 views
Conservation of angular momentum across different reference frames?
I saw the following problem from the USAPhO:
A uniform pool ball of radius $r$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ ...
2
votes
3answers
571 views
Why do rolling disc (coin) move in circular path?
We have a coin that is rolled such that it's tilted at an small angle $ \theta $.
Question:: What turns around rolling disc so that it traces circular motion (spiral as it's speed decreses)?
...
0
votes
0answers
67 views
How to control heavy ball in x and y direction? [closed]
I'm trying to build a device that holds a rather heavy ball and I want to be able to control it's "movement" in the X and Y directions (they are horizontal as well as orthogonal).
My current ...
2
votes
2answers
148 views
Multiplicity of eigenvalues of angular momentum
Reading Dirac's Principles of Quantum Mechanics, I encounter in § 36 (Properties of angular momentum) this fragment:
This is for a dynamical system with two angular momenta $\mathbf{m}_1$ and ...
4
votes
2answers
735 views
Why Silver atoms were used in Stern-Gerlach experiment?
For the Stern-Gerlach experiment done in 1922:
1-why silver atoms were used?
2-Silver atom contains many electrons in different orbits (different $l$'s). Wouldn't the inner -shell electrons be ...
2
votes
1answer
1k views
What's the right way to calculate the principal moment of inertia?
I am writing a program that incorporates calculating the principal moment of inertia for a protein residue based on its component atom XYZ coordinates. I am exceedingly confused about which formulas ...
1
vote
2answers
298 views
motion in the body-fixed frame?
This is really basic, I'm sure: For rigid body motion, Euler's equations refer to $L_i$ and $\omega_i$ as measured in the fixed-body frame. But that frame is just that: fixed in the body. So how ...
7
votes
3answers
282 views
Can one make a ball rotate around a vertical axis using only a combination of horizontal axis rotations?
This is a nice problem that I would like to share.
Problem: In a public garden, there a statue consisting of a spherical stone and a stone cup. The ball is 1 meter in diameter and weighs at least a ...
3
votes
2answers
292 views
How is angular momentum measured in experiments/in practice? [duplicate]
Possible Duplicate:
How does one experimentally determine chirality, helicity and spin?
How do you find spin of a particle from experimental data?
We read about and study angular momentum ...
1
vote
2answers
106 views
Why isn't the maximum eigenvalue of $J_z$ squared equal to the maximum eigenvalue of $J^2$?
During a standard derivation of the eigenvalues of the angular momentum operators, $J^2$ and $J_z$, where
$$J^2|\alpha, \beta\rangle =\hbar^2\alpha|\alpha, \beta\rangle$$
and
$$J_z|\alpha, ...
0
votes
1answer
171 views
Angular momentum conservation in a central field through the Hamiltonian
In my teacher's notes there is a discussion of the Hamiltonian for a central force field with potential $V(r)$.
The Hamiltonian is formulated in spherical polar coordinates:
...
1
vote
2answers
275 views
what does it mean for a particle with no size to have angular momenta?
I recently was reading about higgs boson and particle spin recently and I stubble upon an question that contains an answer to what a spin is.
It explains that electrons etc. have no size yet they ...
2
votes
1answer
396 views
What does it really mean that particle has a spin of up/down? And how is spin actually meassured?
I been reading some physics articles (related to the recent discovery of the particle that could be a Higgs boson) posted online and it was talking about electron spin and how it can only have values ...
2
votes
1answer
192 views
Angular momentum components as independent integrals of motion
I was told that in order to solve the Kepler problem (6 degrees of freedom in total) you have to proceed, step by step, to reduce those degrees of freedom using the integrals of motion. You do so ...
2
votes
1answer
226 views
Double gyroscope: Can a spinning pencil tumble on only one axis?
Picture an object such as item 7 on this page .. http://en.wikipedia.org/wiki/List_of_moment_of_inertia_tensors
Call that the x axis and z is in to the distance. See diagram below.
We are in deep ...
2
votes
3answers
828 views
Proving angular momentum is conserved for a particle moving in a central force field $\vec F =\phi(r) \vec r$
A problem I am trying to work out is as follows:
A particle moves in a force field given by
$\vec F =\phi(r) \vec r$. Prove that the angular momentum of the particle about the origin is constant.
...
5
votes
3answers
397 views
Why does spin have a discrete spectrum?
Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) continuous values, spin has a (probabilistic) discrete spectrum?
7
votes
3answers
89 views
Rotationally invariant body and principal axis
Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$).
Is it true that in this case the ...
1
vote
1answer
276 views
Cases in which angular velocity and angular momentum point into same direction
I know that angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ of a rigid body doesn't point into the same direction in general. However if your body spins around a principal axis, ...
10
votes
3answers
833 views
Adding 3 electron spins
I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
3
votes
1answer
165 views
Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients
An identity exists for CG coefficients:
$$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$
But why is there a phase factor $(-1)^{j_1+j_2-J}$?
It seems ...
2
votes
3answers
197 views
Should any theory of physics respect the principle of conservation of angular momentum or linear momentum?
Is it possible that a theory that can describe the universe at the planck scale can violate things that we now consider fundamental in nature?For example can it violate rotational and translational ...
3
votes
3answers
393 views
$\hbar$, the angular momentum and the action
Is there anything interesting to say about the fact that $\hbar$, the angular momentum and the action have the same units or is it a pure coincidence?
1
vote
1answer
246 views
Wigner-Eckart projection theorem
I'm following the proof of Wigner-Eckart projection theorem which states that:
$$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
8
votes
2answers
361 views
Lie bracket for Lie algebra of $SO(n,m)$
How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by
$$[J_{ab},J_{cd}]
~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$
...
2
votes
1answer
981 views
Angular momentum operator and expectation values
I was reading some notes and it says that $\langle L_z^2\rangle=\langle L^2\rangle$ IFF the system is radially symmetric. I can see that in order that the LHS of the statement implies that $\langle ...
2
votes
2answers
482 views
Conservation of angular momentum in helicopter
I have a small RC-controlled toy helicopter with removable tail rotor.
Suppose I remove the tail rotor, hold the tail with my hand, start the rotor until it moves with constant angular velocity and ...
5
votes
3answers
386 views
Can the spin of a photon change during its “life”?
Or is the spin set in one of two possible states at its moment of creation and does not change for the rest of the duration of its "life"?
4
votes
1answer
432 views
Angular momentum operator in terms of ladder operators
I wanted to show that the angular momentum of the particle state with zero momentum $| \vec{0} \rangle$ is $0$, that is to say the intrinsic spin of a scalar field is $0$ using a mode expansion. There ...
4
votes
1answer
122 views
Eigenvalue of $L_z$
In section 4.3 of Griffths' "Introduction to Quantum Mechanics", just below Figure 4.6, the sentence begins
Let $\hbar \ell$ be the eigenvalue of $L_z$ at this top rung...
Why is this valid? ...
2
votes
1answer
137 views
Will a precessing spinning whell fall down if there is no friction at all?
If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down?
I just asked another question about the same problem:
Direction ...
1
vote
3answers
798 views
Direction of torque precession of a spinning wheel
Consider a spinning wheel, which is held up by one end of it's axis like this:
To explain why the change of angular momentum is directed as shown in the figure above, one usually says that there is ...
4
votes
2answers
126 views
Would the arms of a rotating ice skater still move outwards if there was no other object in the universe?
If there is no other object in the universe apart from a rotating ice skater, then nothing can be used as a reference frame. Would it make any sense to say that the skater is rotating? If so, rotating ...
2
votes
1answer
203 views
How could $\textbf{S}^2$ not be a multiple of the identity?
I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$:
As will be shown in ...
0
votes
1answer
87 views
Period of an Object in Periodic Motion
My attempt (if it matters):
The initial period is given by $T_X = \frac{2\pi X}{v}$ for some $v$.
The new period is given by $T_Y = \frac{2\pi Y}{v}$ for the same $v$.
$Y = \frac{X}{2}$, so ...
6
votes
3answers
852 views
Conservation of angular momentum for a rigid body rotating about a fixed point
Picture a rigid body such as a sledge hammer. Imagine that the base of the handle is attached to a fixed point such that it can rotate but not translate. I give the hammer a good push to get it ...
5
votes
1answer
192 views
Effect of the tail of the cat in the falling cat problem
To explain why a falling cat can turn by 180 degree without external torque and without violation of the conservation of angular momentum, one usually models the cat as two cylinders as in
...
0
votes
1answer
319 views
Probability of getting a particular spin
I'm a beginner in quantum mechanics, and I'm a bit confused about states and the probability to measure certain values. I would like to understand at least the following simplified situation:
...
3
votes
1answer
251 views
Angular Momentum Addition Theorem - Sanity Check
Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as:
$j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation)
, but I'm a little unsure how to ...
1
vote
2answers
286 views
Conservation of angular momentum in propeller planes and helicopters
Consider a propeller plane with only one propeller in the front. If the propeller rotates, I would expect by conservation of angular momentum, that the body of the plane would spin in the opposite ...
3
votes
3answers
534 views
How do the Planets and Sun get their initial rotation?
How do the Planets and Sun get their initial rotation?
Why do Venus and Mercury rotate so slowly compared to other planets and why does Venus rotate in a different direction to Mercury, Earth and ...
4
votes
1answer
329 views
Is all angular momentum quantized?
Angular momentum is definitely quantized in elementary particles and electrons in atoms.
Molecules also have characteristic rotation spectra.
Is it true that all angular momentum is quantized, ...
2
votes
1answer
478 views
General procedure for Clebsch-Gordan expansions
I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
2
votes
2answers
257 views
Why does optical pumping of Rubidium require presence of magnetic field?
The optical pumping experiment of Rubidium requires the presence of magnetic field, but I don't understand why.
The basic principle of pumping is that the selection rule forbids transition from ...
2
votes
3answers
1k views
Angular momentum equations
I do not understand this because angular momentum is $L=I\omega$ ($I$ is moment of inertia;$\omega$ is angular velocity) but it I have also seen equations where $L= rmv\sin(x)$. I do not understand ...
0
votes
1answer
563 views
Derivation of angular momentum commutator relations
I'm trying to understand the derivation of the angular momentum commutator relations. How is
$$[zp_y, zp_x] ~=~ 0?$$
How is
$$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$
3
votes
1answer
524 views
Angular momentum coupling-calculation of Clebsch–Gordan coefficients
I am facing problem in calculating the value of given Clebsch–Gordan coefficients representing the coupled angular momenta of two-particle system. For example
$$\begin{pmatrix}2 & 1 & 2 \\ 1 ...
3
votes
3answers
269 views
The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices
The Pauli spin matrices
$$
\sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
11
votes
3answers
320 views
Do particles have different spins in different frames of reference?
Let's say we have two photons, whose momentum vectors point to opposite directions. Also spin angular momentum vectors of the photons point to opposite directions. (Sum of spins is zero)
Now we ...
